11-Sequences & Summations - EECS 210 Discrete...

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Sequences & Summations David O. Johnson EECS 210 (Fall 2016) 1 EECS 210 Discrete Structures David O. Johnson Fall 2016
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Reminders Homework 3 due: Thursday, October 6 before your lecture Connect 3 due: 11:59 PM, Thursday, October 13 Check BlackBoard for missing grades Grades so far: High-Median-Low Median half above and half below; not the average HW 1: 0-88-100 Connect 1: 0-95-100 HW 2: 0-77-100 Exam 1: 52-95-100 Running: 46-89-99 Sequences & Summations David O. Johnson EECS 210 (Fall 2016) 2
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Any Questions? Sequences & Summations 3 David O. Johnson EECS 210 (Fall 2016)
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Sequences & Summations (Section 2.4) Sequences Geometric Progression Arithmetic Progression Strings Recurrence Relations Solving Recurrence Relations Summations Product Notation Sequences & Summations 4 David O. Johnson EECS 210 (Fall 2016)
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Introduction Sequences are ordered sets. {1, 2, 3, 5, 8} {1, 3, 9, 27, 81, …} Sequences arise throughout mathematics, computer science, and in many other disciplines, ranging from botany to music. We will introduce the terminology to represent sequences and sums of the terms in the sequences. Sequences & Summations 5 David O. Johnson EECS 210 (Fall 2016)
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Sequences Definition : A sequence is a function from a subset of the integers (usually either the set { 0, 1, 2, 3, 4, …..} or { 1, 2, 3, 4, ….} ) to a set S . The notation a n is used to denote the image of the integer n . We can think of a n as the equivalent of f(n) where f is a function from { 0,1,2 ,…..} to S . We call a n a term of the sequence. Sequences & Summations 6 David O. Johnson EECS 210 (Fall 2016)
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Sequences Example : Consider the sequence where Sequences & Summations 7 David O. Johnson EECS 210 (Fall 2016) { a n } = { , }
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Geometric Progression Definition : A geometric progression is a sequence of the form: where the initial term a and the common ratio r are real numbers. Examples : 1. Let a = 1 and r = −1 . Then: 2. Let a = 2 and r = 5 . Then: 3. Let a = 6 and r = 1/3 . Then: Sequences & Summations 8 David O. Johnson EECS 210 (Fall 2016)
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Arithmetic Progression Definition : A arithmetic progression is a sequence of the form: where the initial term a and the common difference d are real numbers. Examples : 1. Let a = 1 and d = 4 : 2. Let a = 7 and d = 3 : 3. Let a = 1 and d = 2 : Sequences & Summations 9 David O. Johnson EECS 210 (Fall 2016)
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Strings Definition : A string is a finite sequence of characters from a finite set (an alphabet). Sequences of characters or bits are important in computer science.
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